Abstract

Thin observation module by bounded optics (TOMBO) is an optical system that achieves compactness and thinness by replacing a conventional large full aperture by a lenslet array with several smaller apertures. This array allows us to collect diverse low-resolution measurements. Finding an efficient way of combining these diverse measurements to make a high-resolution image is an important research problem. We focus on finding a computational method for performing the resolution restoration and evaluating the method via simulations. Our approach is based on advanced signal-processing concepts: we construct a computational data model based on Fourier optics and propose restoration algorithms based on minimization of an information-theoretic measure, called Csiszár’s I divergence between two nonnegative quantities: the measured data and the hypothetical images that are induced by our algorithms through the use of our computational data model. We also incorporate Poisson and Gaussian noise processes to model the physical measurements. To solve the optimization problem, we adapt the popular expectation-maximization method. These iterative algorithms, in a multiplicative form, preserve powerful nonnegativity constraints. We further incorporate a regularization based on minimization of total variation to suppress incurring artifacts such as roughness on the surfaces of the estimates. Two sets of simulation examples show that the algorithms can produce very high-quality estimates from noiseless measurements and reasonably good estimates from noisy measurements, even when the measurements are incomplete. Several interesting and useful avenues for future work such as the effects of measurement selection are suggested in our conclusional remarks.

© 2008 Optical Society of America

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    [CrossRef]

2007

A. Bouhamidi and K. Jbilou, “Sylvester Tikhonov-regularization methods,” J. Comput. Appl. Math. 206, 86-98 (2007).
[CrossRef]

G. K. Chantas, N. P. Galatsanos, and N. A. Woods, “Super-resolution based on fast registration and maximum a posteriori reconstruction,” IEEE Trans. Image Process. 16, 1821-1830 (2007).
[CrossRef]

2006

S. Farsiu, M. Elad, and P. Milanfar, “Multiframe demosaicing and super-resolution of color images,” IEEE Trans. Image Process. 15, 141-159 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52, 4036-4048 (2006).
[CrossRef]

M. Shankar, R. Willett, N. P. Pitsianis, R. Te Kolste, C. Chen, R. Gibbons, and D. J. Brady, “Ultra-thin multiple-channel LWIR imaging systems,” Proc. SPIE. 6294, 629411 (2006).
[CrossRef]

2005

D. L. Donoho and J. Tanner, “Neighborliness of randomly projected simplices in high dimensions,” in Proc. Natl. Acad. Sci. USA 102, 9452-9457 (2005).
[CrossRef] [PubMed]

2004

W. Li and H. Leung, “A maximum likelihood approach for image registration using control point and intensity,” IEEE Trans. Image Process. 13, 1115-1127 (2004).
[CrossRef]

P. L. Combettes and J. C. Pesquet, “Image restoration subject to a total variation constraint,” IEEE Trans. Image Process. 13, 1213-1222 (2004).
[CrossRef]

2003

S. Park, M. Park, and M. G. Kang, “Super-resolution image reconstruction, a technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165-S187 (2003).
[CrossRef]

2002

S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

P. L. Combettes and J. Luo, “An adaptive level set method for nondifferentiable constrained image recovery,” IEEE Trans. Image Process. 11, 1295-1304 (2002).
[CrossRef]

2001

2000

T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM J. Sci. Comput. 22, 503-516 (2000).
[CrossRef]

1999

M. Elad and A. Feuer, “Super-resolution reconstruction of image sequences,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 817-834 (1999).
[CrossRef]

M. Elad and A. Feuer, “super-resolution restoration of an image sequence--adaptive filtering approach,” IEEE Trans. Image Process. 8, 387-395 (1999).
[CrossRef]

T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM J. Sci. Comput. 20, 1964-1977 (1999).
[CrossRef]

R. H. Chan, T. F. Chan, and C. K. Wong, “Cosine transform based preconditioners for total variation deblurring,” IEEE Trans. Image Process. 8, 1472-1478 (1999).
[CrossRef]

1998

T. F. Chan and C. K. Wong, “Total variation blind deconvolution,” IEEE Trans. Image Process. 7, 370-375 (1998).
[CrossRef]

J. B. A. Maintz and M. A. Viergever, “A survey of medical image registration,” Med. Image Anal. 2, 1-36 (1998).
[CrossRef]

1997

M. Elad and A. Feuer, “Restoration of single super-resolution image from several blurred, noisy, and down-sampled measured images,” IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef]

M. R. Banham and A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14, 24-41 (1997).
[CrossRef]

G. H. Golub and U. von Matt, “Generalized cross-validation for large-scale problems,” J. Comput. Graph. Stat. 6, 1-34 (1997).
[CrossRef]

1995

J. A. Fessler and A. O. Hero, “Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417-1429 (1995).
[CrossRef]

1994

J. A. Fessler and A. O. Hero, “Space-alternating generalized expectation-maximization algorithm,” IEEE Trans. Signal Process. 42, 2664-2677 (1994).
[CrossRef]

1993

Y. Vardi and D. Lee, “From image deblurring to optimal investments: Maximum likelihood solutions for positive linear inverse problems,” J. R. Stat. Soc. B 55, 569-612 (1993).

D. L. Snyder, A. M. Hammoud, and R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014-1023 (1993).
[CrossRef] [PubMed]

S. Kim and W.-Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534-539 (1993).
[CrossRef]

1992

D. L. Snyder, T. J. Schulz, and J. A. O'Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143-1150 (1992).
[CrossRef]

L. G. Brown, “A survey of image registration techniques,” ACM Comput. Surv. 24, 325-376 (1992).
[CrossRef]

T. J. Schulz and D. L. Snyder, “Image recovery from correlations,” J. Opt. Soc. Am. A 9, 1266-1272 (1992).
[CrossRef]

1991

M. I. Miller and B. Roysam, “Bayesian image reconstruction for emission tomography incorporating Good's roughness prior on massively parallel processors,” Proc. Natl. Acad. Sci. USA 88, 3223-3227 (1991).
[CrossRef] [PubMed]

I. Csiszar, “Why least squares and maximum entropy--An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2032-2066 (1991).
[CrossRef]

1990

B. W. Silverman, M. C. Jones, J. D. Wilson, and D. W. Nychka, “A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography,” J. R. Stat. Soc. B 52, 271-324 (1990).

P. J. Green, “On use of the EM for penalized likelihood estimation,” J. R. Stat. Soc. B 52, 443-452 (1990).

P. J. Green, “Bayesian reconstruction from emission tomography data using a modified EM algorithm,” IEEE Trans. Med. Imaging 9, 84-93 (1990).
[CrossRef] [PubMed]

S. Kim, N. Bose, and H. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013-1027 (1990).
[CrossRef]

J. Biemond, R. L. Lagendijk, and R. M. Mersereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856-883 (1990).
[CrossRef]

K. Lange, “Convergence of EM image reconstruction algorithms with Gibbs smoothing,” IEEE Trans. Med. Imaging 9, 439-446 (1990).
[CrossRef] [PubMed]

1987

D. L. Snyder, M. I. Miller, L. J. Thomas, and D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imaging MI-6, 228-238 (1987).
[CrossRef]

Y. Bresler and S. J. Merhav, “Recursive image registration with application to motion estimation,” IEEE Trans. Acoust. Speech Signal Process. 35, 70-85 (1987).
[CrossRef]

1982

L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction in positron emission tomography,” IEEE Trans. Med Imaging 1, 113-122 (1982).
[CrossRef] [PubMed]

1977

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1-38 (1977).

1972

1971

I. J. Good and R. A. Gaskins, “Nonparametric roughness penalties for probability densities,” Biometrika 58, 255-277 (1971).
[CrossRef]

1951

S. Kullback and R. A. Leibler, “On information and sufficiency,” Ann. Math. Stat. 22, 79-86 (1951).
[CrossRef]

Ackerman, J. R.

A. V. Kanaev, J. R. Ackerman, E. F. Fleet, and D. A. Scribner, “Compact TOMBO sensor with scene-independent super-resolution processing,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings, OSA Technical Digest (CD) (Optical Society of America, 2007), paper CMA3.
[PubMed]

Baker, S.

S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

Banham, M. R.

M. R. Banham and A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14, 24-41 (1997).
[CrossRef]

Bascle, B.

B. Bascle, A. Blake, and A. Zisserman, “Motion deblurring and super-resolution from an image sequence,” in Computer Vision--ECCV '96 (Springer,1996), pp. 573-581.

Biemond, J.

J. Biemond, R. L. Lagendijk, and R. M. Mersereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856-883 (1990).
[CrossRef]

Blake, A.

B. Bascle, A. Blake, and A. Zisserman, “Motion deblurring and super-resolution from an image sequence,” in Computer Vision--ECCV '96 (Springer,1996), pp. 573-581.

Blomgren, P.

P. Blomgren, T. F. Chan, P. Mulet, and C. K. Wong, “Total variation image restoration: numerical methods and extensions,” in Proceedings of International Conference on Image Processing (IEEE, 1997), pp. 384-387.

Bose, N.

S. Kim, N. Bose, and H. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013-1027 (1990).
[CrossRef]

Bouhamidi, A.

A. Bouhamidi and K. Jbilou, “Sylvester Tikhonov-regularization methods,” J. Comput. Appl. Math. 206, 86-98 (2007).
[CrossRef]

Brady, D. J.

M. Shankar, R. Willett, N. P. Pitsianis, R. Te Kolste, C. Chen, R. Gibbons, and D. J. Brady, “Ultra-thin multiple-channel LWIR imaging systems,” Proc. SPIE. 6294, 629411 (2006).
[CrossRef]

D. J. Brady, M. A. Fiddy, U. Shahid, and T. J. Suleski, “Compressive optical MONTAGE photography initiative: noise and error analysis,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings, OSA Technical Digest (Optical Society of America, 2005), paper CMB3.
[PubMed]

N. P. Pitsianis, D. J. Brady, and X. Sun, “The MONTAGE least gradient image reconstruction,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings, OSA Technical Digest (CD) (Optical Society of America, 2007), paper CTuB3.

Bresler, Y.

Y. Bresler and S. J. Merhav, “Recursive image registration with application to motion estimation,” IEEE Trans. Acoust. Speech Signal Process. 35, 70-85 (1987).
[CrossRef]

Brown, L. G.

L. G. Brown, “A survey of image registration techniques,” ACM Comput. Surv. 24, 325-376 (1992).
[CrossRef]

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

Carter, J.

M. E. Testorf, J. Carter, M. A. Fiddy, and T. J. Suleski, “Multi-aperture diversity imaging: Physical limitations to the generalized sampling theorem (GST),” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings, OSA Technical Digest (CD) (Optical Society of America, 2007), paper CMA5.
[PubMed]

Chan, R. H.

R. H. Chan, T. F. Chan, and C. K. Wong, “Cosine transform based preconditioners for total variation deblurring,” IEEE Trans. Image Process. 8, 1472-1478 (1999).
[CrossRef]

Chan, T.

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19, S165-S187 (2003).
[CrossRef]

T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAM J. Sci. Comput. 22, 503-516 (2000).
[CrossRef]

T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, 1st ed. (Society for Industrial and Applied Mathematics, 2005).

E. Jonsson, S. Huang, and T. Chan, “Total variation regularization in positron emission tomography,” UCLA Computational and Applied Mathematics Rep. 98-48 (U. California Los Angeles, 1998).

Chan, T. F.

R. H. Chan, T. F. Chan, and C. K. Wong, “Cosine transform based preconditioners for total variation deblurring,” IEEE Trans. Image Process. 8, 1472-1478 (1999).
[CrossRef]

T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM J. Sci. Comput. 20, 1964-1977 (1999).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Comparison of (a) a conventional optical system and (b) a TOMBO system.

Fig. 2
Fig. 2

Comparison of spatial-frequency responses of (a) a conventional (full-aperture) optical system and (b) a TOMBO system.

Fig. 3
Fig. 3

Illustration of the focal-plane shifts.

Fig. 4
Fig. 4

Restoration results of the large-pie baseline image from noiseless measurements: (a) a baseline image, (b) an LR COMPI image produced by a lenslet, (c) an unregularized estimate at iteration 500, and (d) a regularized estimate with α = 0.005 at iteration 500. The number of LR COMPI images is nine in this simulation. Since the detector undersample factor is 4, the estimation problem is approximately 2 × undersampled. (Note that all images were normalized so that the baseline image and the estimates have the same total intensities for clear illustration and comparison. This normalization will be used to all simulation result illustrations in this section.).

Fig. 5
Fig. 5

Set of nine subpixel shifts used for the simulation study in this section: The black squares represent the subpixel shifts in a 4 × 4 detector-undersampling filter, and the white squares represent the missing shift positions. Thus, the available data proportion is approximately 56%. The black square on the top-left corner represents no shift [namely, Δ k = ( 0 , 0 ) ], and the black square on the bottom-right corner represents a subpixel shift of the PSF by Δ k = ( 3 , 3 ) [see Eq. (6)].

Fig. 6
Fig. 6

Restoration results of the large-pie baseline image from noisy measurements: (a) a noisy baseline image, (b) a noisy LR COMPI image produced by a lenslet, (c) an unregularized estimate at iteration 300, and (d) a regularized estimate with α = 0.05 at iteration 200. The number of LR COMPI images is nine in this simulation. For the baseline image, the number of the source photons and the background photons were assumed to be 9 × 10 6 and 6.5 × 10 7 , respectively, and the rms of the detector readout noise was assumed to be 20. For the COMPI data, the number of source photons and background photons were assumed to be 1.0 × 10 6 and 8.2 × 10 5 , and the rms of detector readout noise was assumed to be 20. The noise is then generated according to Eq. (9).

Fig. 7
Fig. 7

Restoration results of the small-pie baseline image from noiseless measurements: (a) a baseline image, (b) an LR COMPI image produced by a lenslet, (c) an unregularized estimate at iteration 500, and (d) a regularized estimate with α = 0.005 at iteration 500. The number of LR COMPI images is nine in this simulation. Since the detector undersample factor is 4, the estimation problem is approximately 2 × undersampled. The diameter of the small pie is one half of that of the large pie in Fig. 4.

Fig. 8
Fig. 8

Restoration results of the small-pie baseline image from noisy measurements: (a) a noisy baseline image, (b) a noisy LR COMPI image produced by a lenslet, (c) an unregularized estimate at iteration 250, and (d) a regularized estimate with α = 0.05 at iteration 300. The number of LR COMPI images is nine in this simulation. Photon counts and the noise level are assumed to be the same as those for the simulation in Fig. 6.

Fig. 9
Fig. 9

Plot of rms errors as the error std σ of the shift estimate Δ ^ k changes. Along the x axis, number 0.4 means the error std σ is 40% (about 10 μm ) of a COMPI detector width, and other numbers can be interpreted in the same way. The plot shows the average rms errors of 20 restoration sets.

Fig. 10
Fig. 10

Selected representative regularized restoration estimates of the large-pie baseline image corresponding to error variances (a)  σ = 1 / 25 of a (COMPI) detector width, (b)  σ = 5 / 25 of a detector width, (c)  σ = 10 / 25 of a detector width, and (d)  σ = 25 / 25 of a detector width. The full set of 16 LR COMPI noiseless measurements is used for the simulation.

Equations (21)

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i o ( y ) = h ( y x ) s ( x ) d x ,
h ( y ) = | A ( u ) e j ( 2 π / λ ) y · u d u | 2 ,
i ( n ) = y w n ( y ) i o ( y ) ,
i k ( n ) = y w n ( y ) i o ( y Δ k ) ,
i k ( y ) = y w k ( y , z ) x h k ( z x ) s ( x ) .
h k ( z ) = | A ( u ) e j ( 2 π / λ ) u · Δ k e j ( 2 π / λ ) u · z d u | 2 ,
h k ( z ) = F 1 { F { h } e j ( 2 π / λ ) l · Δ k } ,
i k ( y ) = y w k ( y z ) x h k ( z x ) s ( x ) , i k ( y ) = i k ( y ) | y = P y ,
d ˜ k ( y ) Poisson { a k ( y ) i k ( y ) + b k ( y ) } + z k ( y ) ,
d k ˜ ( y ) + σ k 2 = d k ( y ) Poisson { a k ( y ) i k ( y ) + b k ( y ) + σ k 2 } ,
e k ( y ; s ) = a k ( y ) i k ( y ; s ) + b k ( y ) + σ k 2 ,
D = { d k } k = 1 K , E ( s ) = { e k ( s ) } k = 1 K ,
s ^ = arg min s 0 D ( D , E ( s ) ) ,
D ( D , E ( s ) ) = k y [ d k ( y ) ln d k ( y ) e k ( y ; s ) + e k ( y ; s ) d k ( y ) ] .
s ^ ( j + 1 ) ( x ) = s ^ ( j ) ( x ) k z h k ( z x ) y w ( y z ) d k ( y ) a k ( y ) e k ( y ; s ^ ( j ) ) k z h k ( z x ) y w ( y z ) a k ( y ) .
s ^ = arg min s 0 D ( D , E ( s ) ) + α R TV ( s ) ,
R TV ( s ) = y 1 y 2 [ s ( y 1 , y 2 ) s ( y 1 1 , y 2 ) ] 2 + [ s ( y 1 , y 2 ) s ( y 1 , y 2 1 ) ] 2 ,
s ( x ) D ( D , E ( s ) ) | s = s ^ ( j ) + α s ( x ) R TV ( s ) | s = s ^ ( j ) = 0 ,
s ( x ) D ( D , E ( s ) ) | s = s ^ ( j ) + α s ( x ) R TV ( s ) = 0 ,
s ^ ( j + 1 ) ( x ) = s ^ ( j ) ( x ) k z h k ( z x ) y w ( y z ) d k ( y ) a k ( y ) e k ( y ; s ^ ( j ) ) k z h k ( z x ) y w ( y z ) a k ( y ) + α s ( x ) R TV ( s ) | s = s ^ ( j ) .
Δ k = Δ ^ k + ϵ ,

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